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Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

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Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies Mathematical and Computational Applications Article Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies Mikhail Nikabadze 1,2,∗ and Armine Ulukhanyan 2 1 Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia 2 Department of Computational Mathematics and Mathematical Physics, Bauman Moscow State Technical University, 105005 Moscow, Russia; [email protected] * Correspondence: [email protected] Received: 26 January 2019; Accepted: 21 March 2019; Published: 22 March 2019 Abstract: The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any order and of any even rank is formulated, and also some of its special cases are considered. In particular, using the canonical presentation of the TBM of the tensor of elastic modules of the micropolar theory, in the canonical form the specific deformation energy and the constitutive relations are written. With the help of the introduced TBM operator, the equations of motion of a micropolar arbitrarily anisotropic medium are written, and also the boundary conditions are written down by means of the introduced TBM operator of the stress and the couple stress vectors. The formulations of initial-boundary value problems in these terms for an arbitrary anisotropic medium are given. The questions on the decomposition of initial-boundary value problems of elasticity and thin body theory for some anisotropic media are considered. In particular, the initial-boundary problems of the micropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators (tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transversely isotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators (tensors–tensors) of the initial-boundary value problems are constructed that allow decomposing initial-boundary value problems. We also find the determinant and the tensor of cofactors to the sum of six tensors used for decomposition of initial-boundary value problems. From three-dimensional decomposed initial-boundary value problems, the corresponding decomposed initial-boundary value problems for the theories of thin bodies are obtained. Keywords: tensor–operator of equations; stress tensor–operator; tensor–operator of stress and couple stress; tensor–block matrix operator; canonical presentation of tensor–block matrix; eigenoperator MSC: 15A72; 47A75; 74J05; 74J10; 74B05; 74E10; 74E15; 53A45 1. Introduction For isotropic materials, eigenmodules (eigenvalues) and eigenstates (eigentensors) are known since monograph [1]. For anisotropic ones, the said notions were introduced by Kelvin in the middle of the 19th century (other terms were used). However, the investigation in this area was continued only about 40 years ago (see, e.g., [2–15] and the bibliography of [10]). In [10], Ostrosablin has studied the inner structure rather well, classified anisotropic classically linearly elastic materials, and studied many other important problems (see also [16]). The notion of elastic eigenstates is applied in the plasticity theory (see [17]) and the flow theory (see [18]). Representations of general solutions of the Lamé’s equations were made by many scientists (see, for example, [19–23]), and representations of general solutions of equations in displacements and rotations Math. Comput. Appl. 2019, 24, 33; doi:10.3390/mca24010033 www.mdpi.com/journal/mca Math. Comput. Appl. 2019, 24, 33 2 of 19 of the micropolar theory of elasticity can be found, for example, in [22,24–26]. Note that in this case the equations of the classical and micropolar theory of elasticity are decomposed, but for decomposing the equations, as well as static boundary conditions, the algebraic method turned out to be more efficient. If this method is used, it is advisable to present the equations and static boundary conditions in tensors–operators (tensor–block operators) in the case of classical (micropolar) medium, and then find the tensors–operators (tensor–block operators) of algebraic cofactors for these operators. Of course, the algebraic method can be used for decomposing the static boundary conditions only for bodies with piecewise plane boundaries. These questions are described in some detail in [27,28], and in this work, special attention is paid to the canonical representations of equations and boundary conditions. Note that in [28] some questions from monograph [27], which was published in Russian in the Mechanics and Mathematics faculty of Lomonosov Moscow State University, were presented in English with some clarifications and changes. In particular, the authors presented some questions of tensor calculus; constructed new versions of theories of single-layer and multi-layer elastic thin bodies via the developed method of orthogonal polynomials and also obtained the corresponding decomposed equations of quasistatic problems of classical (micropolar) theory of prismatic bodies with constant thickness in displacements (in displacements and rotations) from the decomposed equations of classical (micropolar) theory of elasticity. The above results and eigenvalue problems for tensor and tensor–block matrices (see [29]) are used in this work for mathematical modeling of the micropolar thin bodies. 2. Statement of Eigenvalue Problem of a Tensor–Block Matrix of Any Even Rank Find all tensor columns U which satisfy equation p M ⊗ U = lU, where l is scalar, and 0 1 0 1 U1 A11 A12 A13 ··· A1m B C B A A A ··· A C B U2 C B 21 22 23 2m C B · C B C U = B C , M = B A31 A32 A33 ··· A3m C , (1) B · C B C @ · A @ ··············· A Um Am1 Am2 Am3 ··· Amm j j ···j 1 2 p i ···ip j i = R 1 R ··· = R R , R ··· = r ⊗ · · · ⊗ rp, M M i1i2···ip j1 jp Mi j i1 ip 1 p i i ···i j j 1 2 p j j2···jp i · j Ri ⊗ R = g , A = A Ri i ···i R 1 = A RiR , i kl kl, j1 j2···jp 1 2 p kl, · j = i1···ip = = Uk Uk, i1i2···ip R , k, l 1, m, i1, i2,..., ip, j1, j2,..., jp 1, n, i, j =1, N, N = np. Note that this problem is solved for the tensor of any even rank and the tensor–block matrix of any even rank consisting of four tensors, as well as for the tensor and the tensor–block matrix of the fourth rank, and published in [29]. Therefore, here we will not dwell on the presentation of this problem with the aim of shortening the letter, but, if necessary, we will refer to the work mentioned in the previous sentence. We also note that, solving the eigenvalue problem for a tensor–block matrix of any even rank consisting of four tensors, there is no difficulty in solving the analogous problem for the tensor–block matrix M (see (1)). Thus, we assume that the eigenvalue problem for a tensor–block matrix of any order and of any even rank is solved and we will consider some its applications below. Math. Comput. Appl. 2019, 24, 33 3 of 19 3. Equations of Motion Relative to the Displacement and Rotation Vectors for an Elastic Material without a Center of Symmetry The constitutive relations (CR) given in [25,30,31] for a linearly elastic inhomogeneous anisotropic material without a center of symmetry for small displacements and rotations and isothermal processes can be written as 2 2 2 2 P = A⊗g+B⊗ m = C⊗g+D⊗ (g = ru−C · j = rj) (2) g {, m g { g ' j, { j , e e e e e e e e e e e e where P and m are the stress and couple-stress tensors, g and { are the tensors of deformation and bending-torsion,e e u and j are the displacement ande rotatione vectors, A, C = BT and D e e e 2 e C e e e ⊗ e are the material tensors of the fourth rank, ' is the discriminant tensor of third rank, is the inner two-product [27,29,32–35], the superscript T in the upper right corner of the quantities denotes transposition. Introducing the tensor columns of the deformation and bending-torsion tensors and stress and couple-stress tensors, as well as the fourth rank tensor–block matrix (TBM) of the elastic modulus tensors ! ! g T P T X = X = g, { , Y = Y = P, m , {e me e e e e e e e e e e ! A B T M = M = M , (3) Ce De e e e e e the specific strain energy and the CR can be written in the form 2 2 2 2 2 2 T 2 2 2 2F(g,{)=g⊗A⊗g+2g⊗B⊗{+{⊗D⊗{ =X ⊗M⊗X, Y=M⊗X. (4) e e e e e e e e e e e e e e e e e If the material has a center of symmetry in the sense of elastic properties, then B = 0, where 0 is the zero tensor of the fourth rank and the tensor–block matrix of the elastic moduluse tensorse (3) wille take the form of a tensor–block-diagonal matrix. Substituting (2) in the equations of motion for small displacements and rotations 2 r · P + F = 2u r · m + C⊗P + m = J 2j r r¶t , m ' r ¶t j, e e e e and introducing the 2nd rank tensor–block matrix operator of the equations of motion and the vector columns of the displacement and rotation vectors and vectors of volume forces and moments ! ! ! A B u rF = = = M e e , U , X , (5) e C D j rm e e we obtain the equations of motion in displacements and rotations in the form M · U + X = 0, (6) e where the differential tensors–operators A, B, C and D have expressions e e e e 0 2 0 ijkl ijkl A = A − Er¶t , A = rjrl(A ri + ri A )rk, e e eijkl e ijkl l ·· mnkj l ·· mnij B = rjrl[(B ri + riB − C· mn A )rk − C· mnri A ], e klij klij j ·· mnkl 0 2 C = rjrl(B ri + riB + C· mn A )rk, D = D − J¶t , (7) e 0 ijkl ijkl j l l j s ·· emnkt e e D = rjrlf[D ri +riD +(gsgt −gsgt)C· mn B ]rk e l ·· pqmn ·· j pqij −C· pq(A Cmn ·+riB )g.
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